[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81843-en":3,"doc-seo-81843-105":31,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},81843,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","From Ham-Sandwich to Centerpoints: Semialgebraic Algorithms for Cutting Polytopal Measures","Exact algorithms are developed for the ham-sandwich theorem and centerpoint results for polytopal measures. The cap-volume function—volume cut off by a halfspace—is shown to be piecewise rational over a natural decomposition of the space of oriented hyperplanes. This reformulates prescribed-proportion cutting as semialgebraic feasibility, giving polynomial-time procedures (fixed dimension) to decide existence, describe all solutions, and sample or enumerate them. The framework extends to the center transversal theorem and yields semialgebraic descriptions of deep affine flats and centerpoints via floating bodies.","arXiv :2607 .02400v 1 [ cs .CG] 2 Jul 2026  \nFrom Ham-Sandwich to Centerpoints: Semialgebraic Algorithms for Cutting Polytopal Measures  \nMarie-Charlotte Brandenburg, Jesús A. De Loera, and Chiara Meroni  \nAbstract  \nWe design exact algorithms for the ham-sandwich and centerpoint theorems for polytopal measures. Our key observation is that the cap-volume function of such a measure, i.e., the volume cut off by a halfspace, is piecewise rational on a natural decomposition of the space of oriented hyperplanes. This lets us recast prescribed-proportion cutting problems as semialgebraic feasibility problems. For fixed ambient dimension, this yields polynomial-time algorithms to decide the existence of cuts, describe the full solution set, and sample or enumerate solutions. We extend this framework to the center transversal theorem, showing that spaces of deep affine flats are semialgebraic, which holds for centerpoints. We further show that the set of centerpoints of a convex polytope coincides with its floating body at level 1/(d + 1), a useful semialgebraic description.  \n1 Introduction  \nIn this article, we develop computational versions of the ham-sandwich theorem [ST42] and the centerpoint theorem [Rad46] . These classical results establish the existence of hyperplanes that partition measures, or point sets, into balanced pieces. Together, they underlie a remarkably rich line of work in computational geometry on efficient equipartitioning, with exciting direct applications to economics, fair-division problems, gerrymandering, and statistics [BT96 ; Su99 ; Sob17 ; FG23 ; FLMN24] .  \nIn this paper, we discuss algorithmic variations of these two classical problems. The difficulty is already visible in the ham-sandwich setting illustrated in Figure 1: given three ingredients, say bread, ham, and cheese, can one cut the sandwich with a single plane so that each ingredient is divided exactly in half? A key contribution of this paper is an algorithm that solves this tasty sandwich problem and finds all possible fair bisections; see Figure 1 .  \nFigure 1: All possible ways to cut this ham-cheese-sandwich into two equal parts.  \nKeywords: ham-sandwich theorem, centerpoint theorem, center transversal theorem, polyhedral geometry, computational real algebraic geometry, floating bodies, equipartitions  \nMSC classes: Primary 52B55; Secondary 52A35, 52A38, 14P10, 14Q30, 68Q25, 68W30 .  \nThe existence of such bisecting cuts follows from famous and elegant proofs, but these proofs are typically non-constructive, relying on tools from algebraic topology or convex geometry. Consequently, many partition results remain mysterious from the point of view of computation. Here we present algorithms for both the ham-sandwich theorem and the centerpoint theorem, as well as for several of their generalizations. Our algorithms apply when the input measures are finite unions of convex polytopes. More precisely:  \nDefinition 1 .1. We call a measure P in Rd polytopal if it is the restriction of the Lebesgue measure to a union of finitely many full-dimensional convex polytopes Q 1 , ... , Qn ⊂ Rd. We refer to Qj as subpolytopes of P. We say that such a measure is rational if the vertices of all Qi have rational coordinates.  \nOur first key observation is that for rational polytopal measures one can compute hamsandwich cuts and centerpoints exactly, using techniques from computational semialgebraic geometry [BPR06 ; BCR98] . For polytopal measures, the sets of bisecting hyperplanes are natural semialgebraic sets, and, in fact, most problems of partitioning polytopes or polytopal measures can also be reframed as feasibility problems in semialgebraic geometry [BPR06] . For all our algorithmic results, the input size includes the number of vertices and the bitsize of their coordinates. Throughout the algorithmic complexity statements, the ambient dimension d is fixed unless explicitly stated otherwise. Outputs are semialgebraic descriptions or real algebraic sam","cbCaiaxPAVbwh5Dj","https://ap.wps.com/l/cbCaiaxPAVbwh5Dj","pdf",1938982,4,1,24,"English","en",105,"# Abstract\n# Introduction\n## Ham-Sandwich Contributions\n## Computational Problem Formulation","[{\"question\":\"What core idea enables the proposed algorithms for cutting polytopal measures?\",\"answer\":\"The cap-volume function, i.e., the volume removed by a halfspace, is piecewise rational on a natural decomposition of the space of oriented hyperplanes. This allows prescribed-proportion cutting to be rewritten as semialgebraic feasibility.\"},{\"question\":\"What computational tasks can be solved in polynomial time for fixed ambient dimension?\",\"answer\":\"The methods decide whether suitable cuts exist, describe the full solution set, and sample or enumerate solutions for the ham-sandwich and related centerpoint problems.\"},{\"question\":\"How are centerpoints characterized using floating bodies?\",\"answer\":\"For a convex polytope, the set of centerpoints coincides with its floating body at level 1/(d+1), providing a useful semialgebraic description.\"}]","From Ham-Sandwich to Centerpoints: Semialgebraic Algorithms for Cutting Polytopal Measures | PDF",1784176596,60,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":29},"from-ham-sandwich-to-centerpoints-semialgebraic-algorithms-for-cutting-polytopal-measures","",{"@graph":37,"@context":86},[38,54,69],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":20},"https://docshare.wps.com/document/from-ham-sandwich-to-centerpoints-semialgebraic-algorithms-for-cutting-polytopal-measures/81843/",{"url":53,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":42,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-29","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What core idea enables the proposed algorithms for cutting polytopal measures?","Question",{"text":76,"@type":77},"The cap-volume function, i.e., the volume removed by a halfspace, is piecewise rational on a natural decomposition of the space of oriented hyperplanes. This allows prescribed-proportion cutting to be rewritten as semialgebraic feasibility.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What computational tasks can be solved in polynomial time for fixed ambient dimension?",{"text":81,"@type":77},"The methods decide whether suitable cuts exist, describe the full solution set, and sample or enumerate solutions for the ham-sandwich and related centerpoint problems.",{"name":83,"@type":74,"acceptedAnswer":84},"How are centerpoints characterized using floating bodies?",{"text":85,"@type":77},"For a convex polytope, the set of centerpoints coincides with its floating body at level 1/(d+1), providing a useful semialgebraic description.","https://schema.org",{"og:url":53,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":47,"category_name":108,"show_sort_weight":30,"slug":109},5,"Comic","comic",{"id":111,"doc_module":4,"doc_module_name":47,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":47,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":47,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":47,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":47,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":47,"category_name":137,"show_sort_weight":107,"slug":138},19,"General","general"]